A hierarchical deep learning and distributionally robust framework for urban multi-energy systems

• An integrated framework combining deep learning-based probabilistic forecasting with hierarchical distributed optimization and distributionally robust optimization for urban multi-energy systems. • A two-level coordination structure enabling scalable management of multiple energy hubs while ensuring global network constraints through an ADMM-based algorithm with adaptive penalty adjustment. • Wasserstein metric-based ambiguity sets provide robust decision-making under forecast uncertainty, achieving 8.7 % cost reduction compared to traditional stochastic programming under distribution shift scenarios. • Computational tractability is enhanced through parallel decomposition and a neural network-based warm-start method that reduces convergence iterations by 51.7 % and total computation time by 51.9 %. • Numerical simulations on a six-hub test system demonstrate 12.3 % operational cost reduction versus deterministic optimization, renewable utilization rates above 95 %, and zero voltage violations across all scenarios. This paper presents a hierarchical optimization framework for distributed energy management in urban multi-energy systems that integrates deep learning-based forecasting with distributionally robust optimization. The research aims to address three critical challenges: managing renewable forecast uncertainties that cause economic losses, enabling scalable coordination without centralized data sharing, and capturing electrical-thermal network interdependencies through combined heat and power units. The proposed approach coordinates multiple interconnected energy hubs, each comprising distributed generation units, combined heat and power systems, energy storage facilities, and demand response resources. A two-level hierarchical structure is developed where a city-level coordinator manages inter-hub energy exchange and network constraints while hub-level operators optimize local resource dispatch. To handle renewable generation and load uncertainties, an ensemble of deep neural networks (including LSTM, CNN, TCN, and attention mechanisms) provides probabilistic forecasts that are incorporated into the optimization framework through Wasserstein metric-based distributionally robust optimization. The coordination between hierarchical levels is achieved using an alternating direction method of multipliers with adaptive penalty parameter adjustment, enabling parallel decomposition of hub-level subproblems while maintaining global consistency. The distributionally robust formulation hedges against ambiguity in forecast error distributions, ensuring solution robustness under distributional shifts without excessive conservatism. A data-driven warm-start strategy using neural networks accelerates the iterative optimization by initializing ADMM with near-optimal solutions learned from historical problem instances. The framework explicitly models linearized electrical network power flow equations and thermal-hydraulic dynamics, capturing the interdependencies between energy carriers. Numerical results on a six-hub urban energy system demonstrate that the proposed method reduces operational costs by 12.3 % compared to deterministic optimization while maintaining computational tractability through parallel decomposition. The distributionally robust approach achieves 8.7 % cost reduction compared to traditional stochastic programming under distribution shift scenarios, validating its effectiveness in managing forecast uncertainty. The framework demonstrates convergence within 15–22 ADMM iterations and achieves 42 % reduction in computational time through the warm-start strategy, confirming its suitability for real-time applications in complex urban energy networks.